For starters, they can come clean. All their press releases have been exercises in trying to say as little as possible, and be as misleading as possible whiile still not literally lying. For example, their non-denial of the $10,000,000 deal with NSA had half the press falsely reporting that RSA claimed there never any $10,000,000 deal.
Dual_EC_DRBG has been documented since 2006/2007 to be an insecure CSPRNG, even without the backdoor. I knew about it for example, and I do not even work in that field. The only way nobody at RSA Security (a huge company specializing in security) could not have heard about it is by putting their hands over their ears and yelling LALALA. And they didn't put 2 and 2 together about why NSA paid them $10,000,000 when the possible backdoor was discussed in the media and the cryptographic community?
I can accept that RSA Security might have been fooled in 2004. But they have not even tried to explain why they kept using Dual_EC_DRBG after 2006/2007. They have been caught with the hand in the cookie jar, and refuse to even try to defend themselves. Why should I try to invent explanations for their innocence for them?
> what evidence could RSA show us that would reinstate our trust
The point is that the circumstantial evidence is so hugely strong. This is not unfair - this is reality.
It is like finding you standing over a corpse in a pool of blood and a knife in your hand, with a $10 million payment to your account from the victims worst enemy. And you refusing to talk about how you got there, or why the victim's worst enemy sent you the $10 million. Do you think I have no right to make assumptions in that case?